Toward the Mechanics of Fractal Materials: Mechanics of Continuum with Fractal Metric

Toward the mechanics of fractal materials: mechanics of continuum with fractal metric Alexander S. Balankin Grupo “Mecánica Fractal”, ESIME-Zacatenco, Instituto Politécnico Nacional, México D.F. 07738, Mexico This paper is devoted to the mechanics of fractal materials. A continuum framework accounting for the topological and metric properties of fractal domains in heterogeneous media is developed. The kinematics of deformations is elucidated and the symmetry of the Cauchy stress tensor is established. The mapping of mechanical problems for fractal materials into the corresponding problems for the fractal continuum is discussed. Stress and strain distributions in elastic fractal bars are analyzed. Some features of acoustic wave propagation and localization in scale-invariant media are briefly discussed. The effect of fractal correlations in the material microstructure on the crack mechanics is revealed. It is shown that the fractal nature of heterogeneity can either delay or assist the crack initiation and propagation, depending on the interplay between metric and topological properties of the fractal domain. PACS: 62.90.+k, 62.20.mm, 61.43.Hv, 89.75.Da 1 I. Introduction Most natural and engineering materials are inherently heterogeneous [1]. The concept of continuum introduces an approximation of real medium by a region of Euclidean space filled by matter with continuous properties, where the term "continuous" refers that the material properties averaged on the length and time scales of interest vary smoothly, except possibly for a finite number of discontinuities. Accordingly, the continuum mechanics comes into play when one examines what is going on inside a body in a smoothed picture that does not go into details about the forces and motions of the sub- scale constituents. In this regard, traditional homogenization methods provide an efficient way to model the mechanical behavior of heterogeneous materials if the length scales are decoupled and the material microstructure has certain translational symmetry [2]. However, (micro-)structures of real heterogeneous materials frequently possess formidably complicated architecture exhibiting statistical scale invariance over many length scales [3,4]. Examples range from gels [5], polymers [6], and biological materials [7] to rocks [8], soils [9], and carbonate reservoirs [10]. For such materials the classical homogenization methods are inapplicable, because heterogeneities play an important role on almost all scales. This is reflected in the material response to external forces [11,12,13,14,15,16]. Hence, mechanics of scale-invariant materials is of tremendous importance for both fundamental and technological interest. In this background, the fractal geometry offers helpful scaling concepts to characterize the scale invariant domains in heterogeneous materials [17,18,19,20]. These include the scale-invariant spatial and size distributions of solid phases and/or defects (e.g. pores or 2 fractures), long-range correlations in the mass (or pore) density distribution, fractal geometry of fracture, pore, and crumpling networks, among others [3-21]. A key advantage of the fractal approach is the possibility to store the data relating to all scales of observation using a relatively small number of parameters that define a structure of greater complexity and rich geometry [22]. Unfortunately, the functions defined on fractals are essentially non-differentiable in the conventional sense [23]. This demands the development of novel tools to deal with fractal materials within a continuum framework. One of them is the concept of non-local fractional derivative [24,25,26,27]. However, the use of non-local fractional calculus implies (reflects) the existence of long-term spatio- temporal memory in the medium [28]. Hence, the non-local fractional calculus may be suitable in cases when the physical nature of this memory is clear, but not in others. In the last cases, one wants to describe the kinematics of deformable fractal media using the local differential operators, despite the existence of long-range correlations in the material structure [29,30,31,32]. In this context, the introduction of differentiable analytic envelopes of non-analytic fractal functions [29] involves, at least implicitly, a continuum approximation of fractal medium. Explicitly, the notion of local fractal continuum was put forward by Tarasov [33]. Further, the fractal continuum approach was employed in Refs. [25,27,31,34,35,36,37,38,39,40,41,42]. The fractal continuum approximation allows us to define the macro properties of heterogeneous materials and express them through the structural parameters. This permits the use of well developed mathematical tools for solving mechanical engineering problems within a continuum framework. 3 However, some fundamental questions regarding to the definition of fractal continuum still remain under debate (see Refs. [43,44]). In the present paper, we put forward a fractal continuum approach accounting for the topological, as well as the metric properties of fractal materials. The paper is organized as follows. Sec. II is devoted to scaling features of fractally heterogeneous materials. Dimension numbers characterizing the scale-invariance, topology, connectivity, and dynamics of fractal medium are outlined. The fractal-continuum homogenization of fractal media is discussed in Sec. III. In this context, the metric, norm, and measure accounting for the scaling properties of heterogeneous materials are introduced. Consequently, the local derivative and generalized Laplacian in the fractal continuum are defied. Sec. IV is devoted to the mechanics of fractal continua. The kinematics of fractal continuum deformations is developed. The Jacobian of transformations is established. Equations of the momentum conservation are derived. Forces and stresses in the fractal continuum are defined. Constitutive laws for fractal continuum are discussed. The mapping of mechanical problems for fractal materials into problems for fractal continua is elucidated in Sec. V. Some specific problems related to mechanics of fractal materials are briefly discussed. Some relevant conclusions are highlighted in Sec. VI. II. Scaling features of fractally heterogeneous materials Generally, a heterogeneous material consists of domains of different materials, or the same material in different phases. Although, in mathematics, fractals can be defined without any reference to the embedding space [45], the natural and engineering materials 4 3 3 reside in the three-dimensional space E and occupy a well defined volume V3 ∈ E . Accordingly, a fractal material necessary consists of fractal and non-fractal domains. For example, if the matrix of porous material is a fractal, the porous space cannot be a fractal and controversially, if the pore space is a fractal, the matrix should be a non-fractal [46]. Furthermore, in both cases the interface between solid matrix and pore space can also be a fractal [47]. The scaling properties of a fractal domain can be characterized by a set of fractional dimensionalities [19,48,49,50]. Most definitions of the fractional dimension numbers are based on the paradigm of domain covering by balls (cubes, tubes, etc.) of some size ε , or at most ε [23,51,52]. In mathematics these covers are considered in the limit ε → 0 and not necessary associated with the scale invariance of studied patterns. At the same time, it was noted that in many cases the number of n-dimensional covers need to cover a fractal of characteristic linear size L scales as N(L / ε ) ∝ ()L / ε D , (1) where ε is the length resolution scale, d < D < n is the fractal (metric or box-counting) dimension, and d is the topological dimension of the fractal pattern, while n is the dimension of the embedding Euclidean space E n [17,18,53,54]. Examples include most classical fractals, such as the Cantor dusts (see Fig. 1), Koch curves, Sierpinski gaskets and carpets (see Fig. 2), Menger sponge (see Fig. 3), and percolation clusters [49,50], among others [17-23]. 5 n Figure 1. Iterative construction of Cantor dusts Φ D = Φαi embedded into: (a) E1 C n ∏i 1 ( D = ln 2/ ln3 ), (b) E 2 (D = ln 4/ ln3 ), and (c) E3 (D = ln8/ ln3 ). Notice that the topological dimension of any Cantor dust is d = 0 , whereas the intrinsic fractal dimension and spectral dimensions are equal to the dimension of the embedding Euclidean space E n , that is 0 d D d d n [64]. = < < l = s = 6 Figure 2. Iterative constructions of 9 fractals of the same topological d = 1 and fractal ( D = ln6/ ln3 < n = 2 ) dimensions, but having different topological and connectivity properties: (a-e) Koch curves; (f) Sierpinski gasket; (g,h) Sierpinski carpets and (i) Cantor circles. The values of spectral dimensions are taken from Refs. [55]. 7 Figure 3. Definitions of the intersection area and the characteristic length scale in the direction normal to the intersection between the Menger sponge ( D = ln 20/ ln3 ) and the Cartesian plane in E3 . The topological dimension of Menger sponge is d =1, whereas the intrinsic fractal dimension is d D n 3 (d 1 ) and the spectral dimension is d 2.5 . The l = < = min = s ≈ intersection of Menger sponge with plane is the Sierpinski carpet of the fractal dimension D ln8/ ln3 1.893, while the intrinsic fractal dimension is d SC D , and so the co- S = ≈ l = S dimension is D D ln 2.5/ ln3 0.83 d / 3 0.91. ζ i = − S = ≈ < γ = l ≈ It is precisely the power-law behavior (1) that allows us to use the powerful tools of fractal geometry for deal with fractal

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